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\author{}
\date{}

\begin{document}

\#! \url{https://zhuanlan.zhihu.com/p/497974257}

\hypertarget{ux968fux673aux53d8ux91cfux53caux5176ux5206ux5e03}{%
\section{随机变量及其分布}\label{ux968fux673aux53d8ux91cfux53caux5176ux5206ux5e03}}

\hypertarget{ux4e00ux968fux673aux53d8ux91cf}{%
\subsection{一、随机变量}\label{ux4e00ux968fux673aux53d8ux91cf}}

\hypertarget{ux5b9aux4e491}{%
\paragraph{定义1}\label{ux5b9aux4e491}}

\begin{quote}
若变量\(X、Y和Z\)都可以看成是定义在随机试验\(E\)的样本空间为\(S = {e}\).上的单值实值函数\(X= X(e)、Y= Y(e)和Z= Z(e)\),我们称其为\textbf{随机变量.}
\end{quote}

\hypertarget{ux4e8cux79bbux6563ux578bux968fux673aux53d8ux91cfux548cux5176ux5206ux5e03}{%
\subsection{二、离散型随机变量和其分布}\label{ux4e8cux79bbux6563ux578bux968fux673aux53d8ux91cfux548cux5176ux5206ux5e03}}

\hypertarget{1-ux79bbux6563ux578bux968fux673aux53d8ux91cf}{%
\subsubsection{1.
离散型随机变量}\label{1-ux79bbux6563ux578bux968fux673aux53d8ux91cf}}

\begin{quote}
如果随机变量\(X\)的所有可能取值是\textbf{有限个}或\textbf{可列无限多}个，则称\(X\)为离散型随机变量
\end{quote}

\begin{itemize}
\item
  \textbf{可列无限多}个，比如\textbf{某地铁站后天的售票数量\(X\)},\textbf{我市110报警中心明天一昼夜收到的呼唤次数\(Z\)};
\end{itemize}

\hypertarget{2-ux5206ux5e03ux5f8b}{%
\subsubsection{2. 分布律}\label{2-ux5206ux5e03ux5f8b}}

\begin{quote}
设离散型随机变量X的所有可能的取值为\(X_k (k=1, 2,\cdots)\),并设\(X\)取各个可能值的概率为\(P{\left\{X=x_{k}\right\}}=p_{k}, \quad k=1,2, \cdots\)则称上式为离散型随机变量\(X\)的分布律(也称概率分布),也可用如下表格表示。
\end{quote}

\begin{figure}
\centering
\includegraphics{https://gitee.com/luobia/note/raw/master/img/image-20220413144418077.png}
\caption{image-20220413144418077}
\end{figure}

\hypertarget{3ux5e38ux89c1ux7684ux79bbux6563ux578bux968fux673aux53d8ux91cfux7684ux5206ux5e03}{%
\subsubsection{3.常见的离散型随机变量的分布}\label{3ux5e38ux89c1ux7684ux79bbux6563ux578bux968fux673aux53d8ux91cfux7684ux5206ux5e03}}

\begin{longtable}[]{@{}llll@{}}
\toprule()
分布 & 介绍 & 分布律 & 记号 \\
\midrule()
\endhead
\(0-1\)分布 & \(X\)只可能取\(01\) &
\(P\{X=k\}=p^{k}(1-p)^{1-k}, \quad k=0,1,\) &
称\(X\)服从参数为\(p\)的\((0-1)\)
分布，简记为随机变量\(X \sim b(1, p)\) \\
二项分布 & \(X \)表示 \(n\) 重伯努利试验中事件 \(A\) 发生的次数 &
\(P\{X=k\}=\mathrm{C}_{n}^{k} p^{k}(1-p)^{n-k}, \quad k=0,1, \cdots, n,\)
&
则称随机变量\(X\)服从参数为\(n, p\)的二项分布，记为\(X\sim b(n, p)\) \\
超几何分布 & \(N\)
件，其中\(M\)件是次品，\textbf{无放回}抽取\(n\)件产品进行检验以\(X\)表示抽取的\(n\)件产品中次品的件数。
&
\(P\{X=k\}=\frac{\mathrm{C}_{M}^{k} \mathrm{C}_{N-M}^{n-k}}{\mathrm{C}_{N}^{n}}, \quad k=min \{n,M\},\)
& 称随机变量X服从超几何分布 \\
泊松分布 & \(X\) 的所有可能取值为
\(0,1,2\cdots\)而取各个值的概率为\(P\{X=k\}\) &
\(P\{X=k\}=\frac{\lambda^{k}}{k !} e^{-\lambda},(k=0,1,2, \cdots ; \lambda>0)\)
& 则称\( X\) 服从参数为\(\lambda\)的泊松分布记为 \(X \sim P(\lambda)\)或
\(X \sim \pi(\lambda)\) \\
& & & \\
\bottomrule()
\end{longtable}

\hypertarget{ux4f2fux52aaux5229ux8bd5ux9a8c}{%
\paragraph{伯努利试验}\label{ux4f2fux52aaux5229ux8bd5ux9a8c}}

\begin{quote}
设随机试验\(E\)只有两个可能的结果，\(A\)及\(\bar{A}\),并且\(P(A)= p, P(A)=1- p=q\),
\((0< p<1)\)将随机试验E独立地重复进行\(n\)次，称这一-串重复的独立试验为\(n\)重\textbf{伯努利试验}.
\end{quote}

\hypertarget{ux4e8cux9879ux5206ux5e03}{%
\paragraph{二项分布}\label{ux4e8cux9879ux5206ux5e03}}

\begin{quote}
\(\sum_{k=0}^{n} P\{X=k\}=\sum_{k=0}^{n} C_{n}^{k} p^{k} q^{n-k}=(p+q)^{n}=1\)
\end{quote}

\begin{quote}
\( \large{\frac{p_{k}}{p_{k-1}}=\frac{C_{n}^{k} p^{k} q^{n-k}}{C_{n}^{k-1} p^{k-1} q^{n-k+1}}=1+\frac{(n+1) p-k}{k(1-p)}},\)

\(So 当  k<(n+1) p  时p_{k}  递增；当  k>(n+1) p  时p_{k}  递减.\)
\end{quote}

\hypertarget{ux6ccaux677eux5206ux5e03ux6ccaux677eux5b9aux7406}{%
\paragraph{泊松分布，泊松定理}\label{ux6ccaux677eux5206ux5e03ux6ccaux677eux5b9aux7406}}

\begin{quote}
\(\text { (2) } \sum_{k=0}^{+\infty} P\{X=k\}=\sum_{k=0}^{+\infty} \frac{\lambda^{k} e^{-\lambda}}{k !}=e^{-\lambda} \sum_{k=0}^{+\infty} \frac{\lambda^{k}}{k !}=e^{-\lambda} e^{\lambda}=1 \text {. }\)
\end{quote}

\begin{quote}
\hl{33}
\end{quote}

\end{document}
